Table Of Content

Foundations of Mechanics Foundatiows sf Mechanics Second Edition Revised, enlarged, and reset A mathematical e~ositbnof classical mechanics with an introduction to the qualitative theory of dynamical systems and appll'cations to the three-body problem RALPHA BRAHAMAN D JERR~LED. M ARSDEN University ofC alifornia Santa Cruz and Berkeley with the assistance of Tudor Ratiu and Richard Cushman Addison-Wesley Publishing Company, lnc. The Advanced Book Program - Redwood City, California Menlo Park, California Reading, Massachusetts - New York Amsterdam Don Mills, Ontario * Sydney Bonn Madrid Singapore Tokyo Bogot6 Santiago San Juan Wokingham, United Kingdom Foundations of Mechanics Second Edition First prinling, 1978 Second printing, with corrections, 1985 Third printing with corrections, 1981 Fourth printing 1982 Fifth printing, with corrections, 1985 Sixth Printing, October 1987 Llbeav of Congress Cetalwlng In Publication Data Abraham, Ralph. Foundations sf mechanics. Bibliography: p. includes index. 1. Mechanics, AnaOflic. 2. Geometry, Differentia!. 3. Mechanics, Celestial. %.M arsden, $errold E., joint author. [I. Title. QAB05.A2 1979 531 '.01'51 77-25858 ISBN 8-8053-010 2-X Copyright @ 1978 by Addison-Wesley Publishing Company. lnc. Published simultaneously in Canada All rights resewed. No part of this publication may be reproduced, stored in a retrieval system, or Branemitbed, in any form or by any means, electronic, mechanical; photocopying, recording or otherwise, without the prior written permision of the publisher. Manufactured in the United States of America 8 9 10-AL-95 94 93 92 91 IN MEMORIAM GLOSSARY OF SYMBOLS E,F, ... finite-dimensional real vector spaces norm of x llxll L (E,P ) continuous linear mapping of E to 6; A' or A*€ L(P,E*) transpose of A E L(E,P ) L (E,F ) multilinear mappings ~ ,(kE, 6 ;) c L (E,F ) skew-symmetric mappings L,k(E,F)cL (E ,F) symmetric mappings U c E open subset f: UcE+F smooth (C" ) mapping x wf (x) effect off on x ~ k f U: C E+L,~(E,F) derivatives off ~ , fU:C E+L,~(E~,F) partial derivatives off - c'(t) = Dc(t) 1 tangent to a curve M,N ,. .. C" manifold n: E+B vector bundle FM(n) C" sections of n TmM tangent space at m EM Tmf or Tf (m) tangent off at m rM: TM+ M tangent bundle 7;: T*cM+ M cotangent bundle (rM):; (MI+ A4 tensor bundles u;: u~(M)+M exterior form bundles f E S(M) C " real-valued functions x E %(MI = vector fields r"(7;) a E % *(M)= one-forms * t E 7; (M)= Tm((rM):) tensor fields @3 tensor product ak rm(&) E (M)= k-forms A exterior product f: M+N mapping of manifolds ak f* : (N)+nk (M) pullback of forms rp: Mc+N diffeomophism oE manifolds rpyr : (MI+ Tsr( N) induced tensor bundle isomorphism rp*: q(M)+T(M) induced tensor field isomorphism U c M open submanifold (U,rp),q: U-U'CE local chart el,...,en basis of E al,. ..,a n dual basis of E* = L(E,R) El>...>En induced generators of %( U) dx', ...,d xn induced dual generators of 5% *( U) F,: q X c M x R + M integral of vector field Lx Lie derivative Ix, Yl Lie bracket exterior derivative inner product volume n form measure of divergence of a vector field determinant of a mapping symplectic form lowering action raising action Hamiltonian vector field symplectic group canonical one-form on T*M canonical two-form on T*M Poisson bracket of functions Poisson bracket of one-forms locally Hamiltonian vector fields globally Hamiltonian vector fields energy surface Legendre transformation pullback of w, by FL symplectic form determined by a metric symplectic manifold action of a Lie group G on P Lie algebra of G momentum mapping dual momentum mapping reduced phase space level surface of N X J amended or effective potential pullback of w to R x M time-dependent vector field vector field associated to X unit time vector field on R X M Cartan form canonical transformation generating function of I; embedding at time t Hamilton principal functions Contents . . . . . . . . . . . . . . . . . . . . . . . . . Preface Bs the Second Edition xili . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface to the First Edition xv Museum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preview mi . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1 Differential Theoy 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Topology 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 17 . . . . . . . . . . . . . . . 1.2 Finite-Dimensional Banach Spaces 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 19 . . . . . . . . . . . . . . . . . . . 1.3 Local Differential Calculus 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 30 . . . . . . . . . . . . . . . . . . . . 1.4 Manifolds and Mappings 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 36 . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Vector Bundles 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 41 . . . . . . . . . . . . . . . . . . . . . . . 1.6 The Tangent Bundle 42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Tensors 52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 59 . . . . . . . . . . . . . . . . . . . . . . Chapter 2 Calculus on Manifolds 60 2.1 Vector Fields as Dynamical Systems . . . . . . . . . . . . . . 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 78 2.2 Vector Fields as Differential Operators . . . . . . . . . . . . .4 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 98 . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Exterior Agebra 101 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 09 2.4 Cartan's Calculus of Differential Forms . . . . . . . . . . . 109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 121 2.5 Orientable Manifolds . . . . . . . . . . . . . . . . . . . . . . 122 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 131 2.4 Integration on Manifolds . . . . . . . . . . . . . . . . . . . . 13 1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 2.7 Some Riemannian Geometry . . . . . . . . . . . . . . . . . .1 44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 156 PART 11 AMALm!CAL DYNAMjCS 159 Chapter 3 . Hamlltonlan end Lagrangian Systems . . . . . . . . . . . .4 61 . . . . . . . . . . . . . . . . . . . . . . . 3.1 Symplectic Algebra 161 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 174 3.2 Symplectic Geometry . . . . . . . . . . . . . . . . . . . . . . 174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 185 3.3 Hamiltonian Vector Fields and Poisson Brackets . . . . . . 187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 199 3.4 Integral Invariants, Energy Surfaces. and Stability . . . . . 201 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 208 . . . . . . . . . . . . . . . . . . . . . . . 3.5 Eagrangian Systems 208 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 17 3.4 The Legendre Transformation . . . . . . . . . . . . . . . . - 218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 223 3.7 Mechanics on Riemannian Manifolds . . . . . . . . . . . . .22 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises -24-4 3.8 Variational Prjlnciples in Mechanics . . . . . . . . . . . . . .24 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 252 Chapter 4 . Hamlltonian Systems with Symmety . . . . . . . . . . . . 253 4.1 Lie Groups and Group Actions . . . . . . . . . . . . . . . . 2 53 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 271 CONTENTS IX 4.2 The Momentum Mapping . . . . . . . . . . . . . . . . . . . 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 295 . . . . . . . . . . 4.3 Reduction of Phase Spaces with Symmetry 298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 309 4.4 Hamiltonian Systems on Lie Groups and the Wigid Body . .3 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 338 4.5 The Topology of Simple Mechanical Systems . . . . . . . . 338 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 359 4.6 The Topology of the Rigid Body . . . . . . . . . . . . . . . .3 60 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 68 Chapter 5 . Hamilton-JacobiT Reor~ga nd Mathematic91 Physics . . . .3 70 5.1 Time-Dependent Systems . . . . . . . . . . . . . . . . . . .3 70 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 378 5.2 Canonical Transformations and Hamilton-Jacobi Theory . 379 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 400 5.3 Eagrangian Submanif o1ds . . . . . . . . . . . . . . . . . . . 402 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 420 5.4 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . .42 5 5.5 Introduction to Infinite-Dimensional Harniltonian System 453 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48 6 5.6 Introduction to Nonlinear Oscillations . . . . . . . . . . . .4 89 PART 1i1 AN OUTLBNE OF OUAhBi$PaB/VED YNAMICS 509 Chapter 6. Topological Dynamics . . . . . . . . . . . . . . . . . . . . . 5 09 . . . . . . . . . . . . . . . . . . . . . 6.1 Limit and Minimal Sets 509 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 513 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Recurrence 513 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 515 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Stability 515 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 519 Chapter 7 . Dltkrentlable Dynamics . . . . . . . . . . . . . . . . . . . .5 20 7.1 Critical Elements . . . . . . . . . . . . . . . . . . . . . . . . 521 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 525 . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Stable Manifolds 525 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 531 7.3 Generic Properties . . . . . . . . . . . . . . . . . . . . . . . .53 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 534 . . . . . . . . . . . . . . . . . . . . . . . 7.4 Structural Stability 534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 536 x CONTENTS 7.5 Pabsolute Stability and h o r n A . . . . . . . . . . . . . . . .5 36 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 7.6 Bifurcations of Generic Arcs . . . . . . . . . . . . . . . . . . 5 43 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 48 7.7 A Zoo of Stable Bifurcations . . . . . . . . . . . . . . . . . .5 48 7.8 Experimental Dynamics . . . . . . . . . . . . . . . . . . . . 570 Chapter 8. Hamiltonsan Dynamics . . . . . . . . . . . . . . . . . . . . -572 Critical Elements . . . . . . . . . . . . . . . . . . . . . . . . 572 Orbit Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . 576 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 Stability of Orbits . . . . . . . . . . . . . . . . . . . . . . . . 579 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 Generic Properties . . . . . . . . . . . . . . . . . . . . . . . .5 87 Structural Stability . . . . . . . . . . . . . . . . . . . . . . . 592 A Zoo of Stable Bifurcations . . . . . . . . . . . . . . . . . .5 95 The General Pathology . . . . . . . . . . . . . . . . . . . . . 6 06 Experimental Mechanics . . . . . . . . . . . . . . . . . . . . 610 PART IV CELESTIAL MECHAMiCS 617 . Chapter 9 The Two-Body Problem . . . . . . . . . . . . . . . . . . . . 619 Models for Two Bodies . . . . . . . . . . . . . . . . . . . . .6 89 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 Elliptic Orbits and Kepler Elements . . . . . . . . . . . . . .6 24 The Delaunay Variables . . . . . . . . . . . . . . . . . . . . 631 Lagrange Brackets of Kepler Elements . . . . . . . . . . . . 635 Wittaker's Method . . . . . . . . . . . . . . . . . . . . . . .6 38 PoincarC Variables . . . . . . . . . . . . . . . . . . . . . . . 647 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 Summary of Models . . . . . . . . . . . . . . . . . . . . . . .6 52 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 Topology of the Two-Body Problem . . . . . . . . . . . . . 656 Chapter 10. The Three-Body Problem . . . . . . . . . . . . . . . . . . . 663 10.1 Models for Three Bodies . . . . . . . . . . . . . . . . . . . . 663 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . -673 10.2 Critical Points in the Restricted Three-Body Problem . . . - 675 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 10.3 Closed Orbits in &ha: Restricted Three-Body Problem . . . . 6 88 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 10.4 Topology of the Planar n-Body Problem . . . . . . . . . . . 699